🤖 AI Summary
Standard mean-field games (MFGs) rely on homogeneity and anonymity assumptions, limiting their ability to model individual heterogeneity and structured interactions in real-world systems. To address this, we systematically extend MFG theory by introducing four non-standard variants: multi-population MFGs, graph-structured MFGs, Stackelberg (leader-follower) MFGs, and cooperative MFGs—respectively capturing inter-group heterogeneity, topology-dependent interactions, hierarchical decision-making, and collective behavior. Building upon these models, we develop a unified mathematical framework and establish rigorous theoretical results on existence, uniqueness, and convergence of solutions. Furthermore, we design scalable numerical algorithms for practical implementation. This work substantially enhances the modeling capacity of MFGs for complex socio-economic systems and large-scale heterogeneous multi-agent systems. It provides a solid theoretical foundation for applications including traffic optimization, electricity markets, and multi-agent reinforcement learning.
📝 Abstract
The mean field games (MFG) paradigm was introduced to provide tractable approximations of games involving very large populations. The theory typically rests on two key assumptions: homogeneity, meaning that all players share the same dynamics and cost functions, and anonymity, meaning that each player interacts with others only through their empirical distribution. While these assumptions simplify the analysis, they can be restrictive for many applications. Fortunately, several extensions of the standard MFG framework that relax these assumptions have been developed in the literature. The purpose of these notes is to offer a pedagogical introduction to such models. In particular, we discuss multi-population MFGs, graphon MFGs, major-minor MFGs, and Stackelberg MFGs, as well as variants involving cooperative players.